Angle projections to Euler angles

Consider a vector in 3D. Its projections on two planes, say YX and YZ planes, makes some angle with the vertical axis ( the y axis in this case). I know these two angles (I call them projected angles). This is the only information I have about the vector.

I need Euler angles which when applied on a unit vector in vertical direction will rotate the unit vector in the direction of the original vector.

Practical application: This would be used to assess angular deformities in tibia/femur. Surgeons know about these projected angles through x-rays in AP and ML planes ( equivalent to YX and YZ planes above). I need to convert them to Euler angles for my application.

If you want to get the Euler angles, you can use solve for cos(theta) = <a,b>/[||a||*||b||] and get an inverse cosine and then put it in the right branch (you will need to also use the cross product where sin(theta) = ||a X a|| / [||a||*||b|| which will allow you to get the quadrant).